Many of the systems in various signal processing applications are nonlinear due to, for example, hardware impairments, such as nonlinear amplifiers and finite-resolution quantization. The Bussgang decomposition is a popular tool used when analyzing the performance of systems that involve such nonlinear components. In a nutshell, the decomposition provides an exact probabilistic relationship between the output and the input of a nonlinearity: the output is equal to a scaled version of the input plus uncorrelated distortion. The decomposition can be used to compute either exact performance results or lower bounds, where the uncorrelated distortion is treated as independent noise. This lecture note explains the basic theory, provides key examples, extends the theory to complex-valued vector signals, and clarifies some potential misconceptions.